direct variation grade9 module 3 by mr. joel garcia

ACTIVITY:
“Agree or
disagree , that is
the question!”

Do you Agree?
 The more time I drive (at a
constant rate), the more distance
I cover.
 If you increase a recipe for more
people, the more of ingredients
you need.
 (In a computer shop)The more
hours you play online games, the
more money you pay.

Do you Agree?
 The more apparels I purchase, the
more money it costs.
 The less time you study, the lower
scores you will get in the exam.
 The less water you drink, the less
trips to the bathroom you have to
make.
 The more time you play temple run,
the longer your cellphone battery
stays.

Direct Variation
What …
Know:__________________
Want to learn: ___________
Had learned:_____________

Definition: y varies
directly as x means that y = kx where y is the
dependent variable, x is the independent
variable and k is the constant of variation.
In other words:
* As x increases in value, y increases or
* As x decreases in value, y decreases.
Direct Variation
Another way of writing this is k =
𝒚
𝒙

Examples of Direct Variation y = kx:
x y
6 12
7 14
8 16
Note: x increases,
6 , 7 , 8
And y increases.
12, 14, 16
What is the constant of variation of the table above?
Since y = kx we can say k =
𝒚
𝒙
Therefore:
12/6=k or k = 2 14/7=k or k = 2
16/8=k or k = 2
EQUATION:
y = 2x
What have you noticed of
the value of k?

x y
10 30
5 15
3 9
Note: x decreases,
10, 5, 3
And y decreases.
30, 15, 9
𝒚
𝒙
30/10 = k or k = 3 15/5=k or k = 3
9/3 = k or k =3
y = 3x is the
equation
Examples of Direct Variation y = kx:
What have you noticed
of the value of k?

Note: x decreases,
-4, -16, -40
And y decreases.
-1, -4, -10
𝒚
𝒙
-1/-4=k or k = ¼ -4/-16=k or k = ¼
-10/-40=k or k = ¼
y = ¼ x is the
equation!
Examples of Direct Variation:
What have you noticed
of the value of k?
x y
-4 -1
-16 -4
-40 -10

What is the constant of variation
for the following direct variation?
Answer
Now
1. 2
2. -2
3. -½
4. ½
x y
4 -8
8 -16
-6 12
3 -6

Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
Yes!
k = 6/4 or 3/2
Equation?
y = 3/2 x
x y
4 6
8 12
12 18
18 27

Yes!
k = 25/10 or 5/2
k = 10/4 or 5/2
Equation?
y = 5/2 x
x y
10 25
6 15
4 10
2 5

X Y
15 5
3 26
1 75
2 150
No!
The k values are
different!

Which of the following is a direct
variation?
1. A
2. B
3. C
4. D
Answer
Now

Which equation describes the
following table of values?
1. y = -2x
2. y = 2x
3. y = ½ x
4. xy = 200
Answer
Now
x y
10 5
2 1
12 6
20 10

Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 28 when
x=7, Find x when y = 52. HOW???
2 -step process
x y
7 28
? 52
1. Find the constant variation
k = y/x or k = 28/7 = 4
k = 4
2. Use y = kx. Find the unknown (x).
52 = 4x or 52/4 = x
x = 13
Therefore:
x =13 when y = 52

x = 9, Find y when x = 40.5. HOW???
2 - step process x y
9 3
40.5 ?
1. Find the constant variation.
k = y/x or k = 3/9 = 1/3
k = 1/3
y = (1/3)40.5
y= 13.5
Therefore:
x = 40.5 when y = 13.5

x = -5, Find y when x = - 8. HOW???
2 - step process
x y
-5 6
-8 ?
1. Find the constant variation.
k = y/x or k = 6/-5 = -1.2
k = -1.2
y= -1.2(-8)
x= 9.6
Therefore:
x = -8 when y = 9.6

Using Direct Variation
to solve word problems
Problem:
A car uses 8 liters of gasoline
to travel 160 km. How much
gasoline will the car use to
travel 400 km?
Step One:
Find points in table
x (gas) y (km)
8 160
? 400
Step Two: Find the constant
variation and equation:
k = y/x or k = 160/8 or 20
Equation: y = 20 x
Step Three: Use the equation
to find the unknown.
400 = 20x
400 = 20x
20 20
or x = 20 liters

Step One:
Find points in table
Alternative Solution:
Step Three:
Solve for the unknown
160
8
=
400
𝑥
160x = 8(400)
or 20 lit.𝑥 =
8(400)
160
Problem: A car uses 8 liters of
gasoline to travel 160 km.
How much gasoline will the
car use to travel 400 km?
x (gas) y (km)
8 160
? 400
Where: x1 = 8, y1 = 160
x2 = ? y2 = 400
Step Two: Form a proportion
Since k1 = k2
𝑦1
𝑥1
=
𝑦2
𝑥2

Step One: Find points in table.
Step Two: Find the constant
variation.
k =
𝑦
𝑥
k =
1000
5
= 200
Step Three:
Use the equation to find the unknown
y = k(x)
y = 200(30) or y = 6000
Problem:
Julio’s wages vary
directly as the number of hours
that he works. If his wag for 5
hours is P1000, how much will
there be in 30 hours?
X (hours) Y (wages)
5 1000
30 ?

Problem:
Julio’s wages vary directly as the number
of hours that he works. If his wage for
5 hours is P1000, how much will there be
in 30 hours?
Use the proportion and solve for the
unknown:
Alternative Method
or
𝑥1
𝑦1
=
𝑥2
𝑦2

EXERCISES: Work as a group, evaluate
and present your answers on the board.
Refer to ACTIVITY 6: Learner’s Manual,
pp. 200 - 202
GROUP 1: A. 1- 2, B. 1, C. 1-2, D. 1
GROUP 2: A. 3 - 4, B. 2, C. 3 - 4, D. 2
GROUP 3: A. 5 - 6, B. 3, C. 5 - 6, D. 3
GROUP 4: A. 7 - 8, B. 4, C. 7 - 8, D. 4
GROUP 5: A. 9 - 10, B. 5, C. 9 - 10, D. 5

Reflect:
How did you find the activity?
What were the problems encountered
in working with the group activity?
How were you able to manage and
mitigate the circumstances you’ve
encountered?

ASSIGNMENT:
A. Choose and evaluate 3 odd problems if your first
name starts with a vowel, otherwise, choose 3 even
numbers if your first name starts with consonant.
Reference: LM, p. 203
B. Make a narrative of your inspiring experience
where knowledge of direct proportion guided and
molded you to be a better individual.
Be ready to share next meeting.

direct variation grade9 module 3 by mr. joel garcia

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direct variation grade9 module 3 by mr. joel garcia